MATHEMATICS REVISION OF FORMULAE AND RESULTS

Surds

a × b = ab

a

b =

a

b

( a)2 = a

Absolute Value

a = a if a ≥ 0

a = − a if a < 0

Geometrically:

x is the distance of x from the origin on the number line

x − y is the distance between x and y on the number line

ab = a . b

a + b ≤ a + b Factorisation

x3 − y3 = x − y (x2 + xy + y2)

x3 + y3= x + y (x2 − xy + y2) Real Functions

A function is even if f −x = f(x). The graph is symmetrical about the y-axis.

A function is odd if f −x = − f(x). The graph has point symmetry about the origin.

The Circle The equation of a circle with:

Centre the origin (0, 0) and radius r units is:

x2 + y2 = r2

Centre (a, b) and radius r units is:

(x − a)2 + (y − b)

2 = r2

Co-ordinate Geometry

Distance formula: d = x2 − x1 2 + (y

2− y

1)2

Gradient formula: m = y2− y1

x2− x1 or m = tanθ

Midpoint Formula: midpoint = x1+ x2

2,

y1+ y2

2

Perpendicular distance from a point to a line:

ax1 + by

1 + c

a2 + b2

Acute angle between two lines (or tangents)

tanθ = m1 − m2

1 + m1m2

Equations of a Line

gradient-intercept form: y = mx + b

point-gradient form: y − y1= m(x − x1)

two point formula: y − y1

x − x1 =

y2− y1

x2− x1

intercept formula: x

a +

y

b = 1

general equation: ax + by + c = 0

Parallel lines: m1 = m2

Perpendicular lines: m1.m2 = − 1

Trigonometric Results

sinθ = opposite

hypotenuse (SOH)

cosθ = adjacent

hypotenuse (CAH)

tanθ = opposite

adjacent (TOA)

Complementary ratios:

sin 90° − θ = cosθ

cos 90° − θ = sinθ

tan 90° − θ = cotθ

sec 90° − θ = cosecθ

cosec(90° − θ) = secθ

Pythagorean Identities

sin2θ + cos2θ = 1

1 + cot2θ = cosec2θ

tan2θ + 1 = sec2θ

tanθ = sinθ

cosθ and cotθ =

cosθ

sinθ

The Sine Rule

a

sinA =

b

sinB =

c

sinC

The Cosine Rule

a2 = b2 + c2 − 2bcCosA

CosA = b

2 + c2 − a2

2bc

The Area of a Triangle

Area = 1

2abSinC

The Quadratic Polynomial

The general quadratics is: y = ax2 + bx + c

The quadratic formula is: x = −b ± b

2− 4ac

2a

The discriminant is: Δ = b2 − 4ac

If Δ ≥ 0 the roots are real

If Δ < 0 the roots are not real

If Δ = 0 the roots are equal

If Δ is a perfect square, the roots are rational

If α and β are the roots of the quadratic equation

ax2 + bx + c = 0

then: α + β = −b

a and αβ =

c

a

The axis of symmetry is: x = −b

2a

If a quadratic function is positive for all values of x, it is

positive definite i.e. Δ < 0 and a > 0

If a quadratic function is negative for all values of x, it

is negative definite i.e. Δ < 0 and a < 0

If a function is sometimes positive and sometimes

negative, it is indefinite i.e. Δ > 0 The Parabola

The parabola x2 = 4ay has vertex (0,0), focus (0,a),

focal length ‘a’ units and directrix y = − a

The parabola (x − h)2 = 4a(y − k) has vertex (h, k)

Differentiation

First Principles:

f ' (x) = lim

h → ∞

f (x + h) – f (x)

h or

f ' (c) = lim

x → c

f (x) – f (c)

h

If y = xn then dy

dx = nxn−1

Chain Rule: d

dx f (u) = f ' (u)

du

dx

Product Rule: If y = uv then dy

dx = u

dv

dx + v

du

dx

Quotient Rule: If y = u

v then

dy

dx =

v du

dx + u

dv

dx

v2

Trigonometric Functions:

d

dx sinx = cosx

d

dx cosx = − sinx

d

dx tanx = sec2x

Exponential Functions: d

dx ef (x) = f ' (x)ef (x)

d

dx ax = ax.lna

Logarithmic Functions: d

dx log

e f (x) =

f ' (x)

f (x)

Geometrical Applications of Differentiation

Stationary points: dy

dx = 0

Increasing function: dy

dx > 0

Decreasing function: dy

dx < 0

Concave up: d

2y

dx2 < 0

Concave down: d

2y

dx2 > 0

Minimum turning point: dy

dx = 0 and

d2y

dx2 > 0

Maximum turning point: dy

dx = 0 and

d2y

dx2 < 0

Points of inflexion: d

2y

dx2= 0 and concavity changes

about the point.

Horizontal points of inflexion: dy

dx = 0 and

d2y

dx2 = 0 and

concavity changes about the point.

Approximation Methods

The Trapezoidal Rule:

f x dx = h

2 y

0+ y

n + 2 y

1+ y

2+ y

3+ …+ y

n−1

b

a

Simpson’s Rule:

f x dx = h

3 y

0+ y

n + 4 y

1+ y

3+ … + 2 y

2+ y

4+ …

b

a

In both rules, h = b − a

n where n is the number of strips.

Integration

If f (x) ≥ 0 for a ≤ x ≤ b, the area bounded by the

curve y = f (x), the x-axis and x = a and x = b is given

by f x dxb

a.

The volume obtained by rotating the curve y = f (x) about the x-axis between x = a and x = b is given by

π f x 2b

a

If dx

dx = xn then y =

xn+1

n + 1

If dx

dx = ax + b n then y =

ax + b n

a(n + 1)

Trigonometric Functions:

sin x dx = − cosx + C

cos x dx = sinx + C

sec2 x dx = tanx + C

Exponential Functions:

eax dx = eax

a+ C and ax dx =

1

ln a.ax

Logarithmic Functions:

f ' (x)

f (x)dx = log

ex + C

Sequences and Series

Arithmetic Progression

d = U2 − U1

Un = a + n − 1 d

Sn = n

2[2a + n − 1 d]

Sn = n

2[a + l] where l is the last term

Geometric Progression

r = U2

U1

Un = arn−1

Sn = a rn− 1

𝑟 − 1=

a 1 − rn

1 − r

S∞= a

1 − r

The Trigonometric Functions

radians = 180⁰

Length of an arc: l = rθ

Area of a sector: A = 1

2r2θ

Area of a segment: A = 1

2r2(θ − sinθ)

[In these formulae, is measured in radians.]

Small angle results:

sinx → 0

cosx → 1

tanx → 0

lim

x → 0 sinx

x = lim

x → 0 tanx

x = 1

For y = sin nx and y = cos nx the period is 2π

n

For y = sin nx the period is π

n

Logarithmic and Exponential Functions

The Index Laws:

ax × ay = ax+y

ax ÷ ay = ax−y

ax y = axy

a−x= 1

ax

ax

y = axy

a0 = 1

The logarithmic Laws:

If loga

b = c then ac = b

loga

x + loga

y = loga

xy

loga

x − loga

y = loga x

y

loga𝑥n + nlog

ax

loga

a = 1 and loga

1 = 0

The Change of Base Result:

loga

b = loge b

loge a=

log10 b

log10 a

EXTENSION 1 REVISION OF FORMULAE AND RESULTS

Co-ordinate Geometry

Dividing an interval in the ratio m:n

mx2 + nx1

m + n,my

2 + ny

1

m + n

Acute angle between two lines (or tangents)

tanθ = m1 − m2

1 + m1m2

Trigonometric Ratios

Sum and Difference Results

sin(A + B) = sinA cosB + cosA sinB

sin(A – B) = sinA cosB – cosA sinB

cos(A + B) = cosA cosB – sinA sinB

cos(A – B) = cosA cosB + sinA sinB

tan(A + B) = tanA + tanB

1– tanAtanB

tan(A – B) = tanA – tanB

1+ tanAtanB

Double Angle Results

sin2A = 2sinA cosA

cos2A = cos2A – sin

2A

cos2A = 1 – 2sin2A

cos2A = 2cos2A – 1

tan2A = 2tanA

1 – tan2A

The ‘t’ Formulae where t = tanθ

2

sin x = 2t

1+ t2

cos x = 1 − t2

1+ t2

tan x = 2t

1 − t2

Subsidiary Angle Method (Rsin(θ + α))

When solving asinθ + bcosθ = c we can

solve by writing in the form Rsin(θ + α) = c

where:

R = a2 + b2 and tan α = b

a

Parameters

The parametric equations for the parabola

x2 = 4ay are x = 2at and y = at

2

All other formulae in this subject are not to be

committed to memory but students must know

how they are derived.

Polynomials

A real polynomial is in the form:

P(x) = pnx2 + pn-1x

n-1 + ... p2x

2 + p1x + p0

p1, p2, p3, ....., pn are coefficients and are real

numbers, usually integers.

The degree of the polynomial is the highest power

of x with non-zero coefficient.

A polynomial of degree n has at most n real roots

but may have less.

The result of a long division can be written in the

form P(x) = A(x) . Q(x) + R(x)

The remainder theorem states that when P(x) is

divided by (x – a) the remainder is P(a).

The factor theorem states that if x = a is a factor

of P(x) then P(a) = 0.

If , , , , ... are the roots of a polynomial then

= −b

a, =

c

a, = −

d

a, =

e

a

Numerical Estimation of the Roots of an Equation

Halving the Interval Method

Newton’s Method

If x = x0 is an approximation to a root of

P(x) = 0 then x1 = x0 – P(x0)

P' (x0) is generally a

better approximation.

Be familiar with the conditions under which

this method fails.

Mathematical Induction

Step 1: Prove result true for n = 1 (It is

sometimes necessary to have a

different first step.)

Step 2: Assume it is true for n = k and then

prove true for n = k + 1

Step 3: Conclusion as given in class

Integration

sin2θ dθ and cos2θ dθ can be solve using the

substitutions:

sin2 =

1

2 1 − cos2θ

cos2 =

1

2 1 + cos2θ

Integration by first making a substitution.

Table of Standard Integrals as provided in HSC

Inverse Trigonometric Functions

y = sin-1

x Domain: –1 x 1

Range: −π

2 ≤ y ≤ −

π

2

y = cos-1

x Domain: –1 x 1

Range: 0 ≤ y ≤

y = tan-1

x Domain: all real x

Range: −π

2 ≤ y ≤ −

π

2

Properties:

sin-1

(-x) = -sin-1

x

cos-1

(-x) = – cos-1

x

tan-1

(-x) = –tan-1

x

sin-1

x + cos-1

x = π

2

sin(sin-1

x) = x

cos(cos-1

x) = x

tan(tan-1

x) = x

General Solutions of Trigonometric Equations:

if sin = q, then = n + (–1)n sin

-1q

if cos = q, then = 2n cos-1

q

if tan = q, then = n + tan-1

q

Derivatives:

d

dx sin

-1x =

1

1− x2

d

dx cos-1x =

−1

1− x2

d

dx tan-1x =

1

1 + x2

Integrals:

1

a2 – x2𝑑𝑥 = sin

-1 x

a = −cos-1

x

a

1

a2 + x2𝑑𝑥 =

1

atan-1

x

a